The vertex of the parabola of an equation of the form   has -coordinate . Here, we substitute , to obtain -coordinate

.

To find the -coordinate, substitute this for :

The vertex is .

To find the -intercepts of the parabola, substitute 0 for  in the equation:

Either

, in which case ,

or

, in which case .

The -intercepts are  and .

The line includes  and either  or , so we find the slope in each case using the slope formula.

If the line includes  and :

.

If the line includes  and :

We choose the first case, since the line has positive slope. The line through  and  has as its equation, using the point-slope form with  and point :

The -intercept of the line coincides with that of the graph of the quadratic equation, which is a horizontal parabola; to find the -intercept of the parabola, substitute 0 for  in the quadratic equation:

The -intercept of the parabola, and of the line, is .

The -intercept of the line coincides with one of those of the parabola; to find the -intercepts of the parabola, substitute 0 for  in the equation:

Either

, in which case ,

or

, in which case .

The -intercepts of the parabola are  and , so the -intercept of the line is one of these. We will examine both possibilities.

If  and  be the - and -intercepts, respectively, of the line, then the slope of the line is . If the intercepts of the line are  and , the slope of the line is ; if the intercepts are  and , the slope is . We choose the latter, since we are looking for a line with positive slope; since its -intercept is , then we can substitute  in the slope-intercept form of the line, , to get the correct equation, .

The point of intersection of the two lines has as its coordinates the values of  and  that make both of the given linear equations true. Therefore, we seek to find the solution of the system of equations:

We need only find , so multiply both sides of the two equations by 7 and 4, respectively. Then add:

,

making 9 the correct response.

The - and -intercepts of the line are, respectively,  and .  If  and  are the - and -intercepts, respectively, of a line, the slope of the line is . This makes the slope of the green line . 

Any line perpendicular to this line must have as its slope the opposite of the reciprocal of this, or . Since the desired line must also have -intercept , then the slope-intercept form of the line is

which can be rewritten in standard form:

The vertices of the triangle are the point of intersection of the graphs of the lines of the two equations, and the -intercepts of those lines.

The -intercept of the line of the equation  can be found by setting  and solving for :

The -intercept of the line of the equation  can be found the same way:

The -intercepts are  and ; these are two of the vertices of the triangle, and since the segment connecting them is horizontal, this will be taken as the base. The length of the base is the difference of the -coordinates:

The intersection of the lines of the equations  and  can be found by solving the system of linear equations, as follows:

The point of intersection, and the third vertex of the triangle, is .

Since we are taking the horiztonal segment to be the base, the height will be the vertical distance to this third point - namely, the -coordinate . The area is half the product of the base and the height:

The intercepts of the line of the equation  can be found by substituting 0 for each variable, in turn:

-intercept:

The -intercept is the point 

-intercept:

The -intercept is the point .

This line and the axes together form a right triangle. The horizontal leg is the segment connecting the origin to , and its length is . The vertical leg is the segment connecting the origin to , and its length is . The area of a right triangle is half the product of these legs, which is 

.

Of the five responses, 45 comes closest to the correct area.